Problem: The arithmetic sequence $a_i$ is defined by the formula: $a_1 = 5$ $a_i = a_{i - 1} + 2$ Find the sum of the first $700$ terms in the sequence.
Getting started Let's write out the first few terms of the series: $5 + 7 + 9 + 11...$ We're dealing with an arithmetic series because the difference between terms is constant. That is, each term is $2$ greater than the one before it. We need a formula to compute the sum of the terms. Formula for arithmetic series The sum $S_n$ of a finite arithmetic series is $S_n = \dfrac {\left(a_1 + a_n \right)}{2} \cdot n$ where $a_1$ is the first term, $a_n$ is the last term, and $n$ is the number of terms. What do we need to use the formula? The first term $(a_1 = 5)$ and the number of terms $(n = {700})$ are given in the question. We need to find the last term $(a_n)$. Step 1: Find $a_n$ (the last term) There are $700 -1= 699$ terms after the first term. The sequence increases by $2$ for each new term. So, the sequence increases by a total of $699 \cdot 2 = 1398$ from where it starts at $5$. That means the last term must be $1398 + 5 = {1403}$. In other words, $a_n = {1403}$. Step 2: Find the sum $(S_n)$ of the series $\begin{aligned} S_n &= \dfrac {\left(a_1 + a_n \right)}{2} \cdot n \\\\ S_{{700}} &= \dfrac {({5} + {1403})}2 \cdot {700} \\\\ S_{700} &= 704(700) \\\\ S_{700} & = 492{,}800 \end{aligned}$ The answer $492{,}800$